Non-generating Partitions of Unimodular Maps
Oliver Pfante and Jürgen Jost
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Submission date: 26. Jan. 2015
MSC-Numbers: 70-XX, 37-XX, 37B10
Keywords and phrases: chaotic dynamics, information theory, aggregation
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We quantify the relationship between the dynamics of a time-discrete dynamical system, driven by a unimodular map T : [0,1] → [0,1] on the unit interval and its iterations Tm, and the induced dynamics at a symbolic level in information theoretical terms. The symbolic dynamics are obtained by a threshold crossing technique. A binary string s of length m is obtained by choosing a partition point α ∈ [0,1] and putting si = 1 or 0 depending on whether Ti(x) is larger or smaller than α. First, we investigate how the choice of the partition point α determines which symbolic sequences are forbidden, that is, cannot occur in the symbolic dynamics. The periodic points of T mark the choices of α where the set of those forbidden sequences changes. Second, we interpret the original dynamics and the symbolic ones as different levels of a complex system. This allows us to quantitatively evaluate a closure measure that has been proposed for identifying emergent macro-levels of a dynamical system. In particular, we see that this measure necessarily has its local minima at those choices of α where also the set of forbidden sequences changes. Third, we study the limit case of infinite binary strings and interpret them as a series of coin tosses. These coin tosses are not i.i.d. but exhibit memory effects which depend on α and can be quantified in terms of the closure measure.